Докажите тождество:
1)

(a + b + c)^{3} - a^{3} - b^{3} - c^{3} = 3 (a + b) (b + c) (a + c)

;
2)

(a - b)^{3} + (b - c)^{3} - (a - c)^{3} = - 3 (a - b) (b - c) (a - c)

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Алгебре 7 класс Мерзляк. Номер №731

(a + b + c)^{3} - a^{3} - b^{3} - c^{3} = 3 (a + b) (b + c) (a + c)

(a + b + c)^{3} - a^{3} - b^{3} - c^{3} = ((a + b + c)^{3} - a^{3}) - (b^{3} + c^{3}) = (a + b + c - a) ((a + b + c)^{2} + a (a + b + c) + a^{2}) - (b + c) (b^{2} - b c + c^{2}) = (b + c) (a^{2} + b^{2} + c^{2} + 2 a b + 2 a c + 2 b c + a^{2} + a b + a c + a^{2} - b^{2} + b c - c^{2}) = (b + c) (3 a^{2} + 3 a b + 3 a c + 3 b c) = 3 (b + c) (a^{2} + a b + a c + b c) = 3 (b + c) ((a^{2} + a b) + (a c + b c)) = 3 (b + c) (a (a + b) + c (a + b)) = 3 (b + c) (a + b) (a + c) = 3 (a + b) (b + c) (a + c)

(a - b)^{3} + (b - c)^{3} - (a - c)^{3} = - 3 (a - b) (b - c) (a - c)

(a - b)^{3} + (b - c)^{3} - (a - c)^{3} = ((a - b)^{3} + (b - c)^{3}) - (a - c)^{3} = (a - b + b - c) ((a - b)^{2} - (a - b) (b - c) + (b - c)^{2}) - (a - c)^{3} = (a - c) (a^{2} - 2 a b + b^{2} - (a b - b^{2} - a c + b c) + b^{2} - 2 b c + c^{2}) - (a - c)^{3} = (a - c) (a^{2} - 2 a b + b^{2} - a b + b^{2} + a c - b c + b^{2} - 2 b c + c^{2}) - (a - c)^{3} = (a - c) (a^{2} - 3 a b + 3 b^{2} - 3 b c + a c + c^{2}) - (a - c)^{3} = (a - c) (a^{2} - 3 a b + 3 b^{2} - 3 b c + a c + c^{2} - (a - c)^{2}) = (a - c) (a^{2} - 3 a b + 3 b^{2} - 3 b c + a c + c^{2} - (a^{2} - 2 a c + c^{2})) = (a - c) (a^{2} - 3 a b + 3 b^{2} - 3 b c + a c + c^{2} - a^{2} + 2 a c - c^{2}) = (a - c) (- 3 a b + 3 b^{2} - 3 b c + 3 a c) = - 3 (a - c) (a b - b^{2} + b c - a c) = - 3 (a - c) ((a b - b^{2}) + (b c - a c)) = - 3 (a - c) (b (a - b) + c (b - a)) = - 3 (a - c) (b (a - b) - c (a - b)) = - 3 (a - c) (a - b) (b - c) = - 3 (a - b) (b - c) (a - c)

ГДЗ Алгебра 7 класс Мерзляк, Полонский, Якир